I don't recall anything in Tolkien's letters about the actual Olympics of the 20th century. In his fiction, the aristocracy of Gondor seems to have had martial skills competitions (the basis for the Olympic Games) or some medieval equivalent like jousting, to judge from this one obscure line in the appendices:

[King] Earnur...was a man of strong body and hot mood; but he would take no wife, for his only pleasure was in fighting, or in exercise of arms. His prowess was such that none in Gondor could stand against him in those weapon-sports in which he delighted, seeming rather a champion than a captain or king... (LotR, App. A.I.iv., "Gondor and the Heirs of Anarion"; bold by squire)

You may remember that Boromir, of the Fellowship, was said to be like Earnur in this regard - but there is no mention then of any games, rather he delighted "chiefly in arms". Assuming that Tolkien was not just compressing the reference, possibly he means to imply that by the end of the Third Age there were no more "weapon-sports" because there was plenty of actual warfare for Boromir and the other captains!

Going further back, the athlete-god of the Valar in the Silmarillion is Tulkas. "He delights in wrestling and in contests of strength; and he rides no steed, for he can outrun all things that go on feet, and he is tireless." (Sil, Valaquenta) Since the Elves and Valar share so many cultural customs, one wonders if the Elves also indulged in such contests.

But I think team Misty Mountain are favourite to win that one. Though the Morgul Trolls look like providing tough competition this year. Also using Dwarves as footballs had been ruled definitely illegal by the meoc. Not a bad sport, all in all but I do wonder why the players feel the urge to hug each other after every point.

Galadriel = Frank Costanza?For instance, on the planet Earth, man had always assumed that he was more intelligent than dolphins because he had achieved so much -- the wheel, New York, wars, and so on -- while all the dolphins had ever done was muck about in the water having a good time. But conversely, the dolphins had always believed that they were far more intelligent than man, for precisely the same reasons. -- Douglas Adams